13th International Conference on Membrane Computing - MTA Sztaki

B. Aman, G. Ciobanu
{
w i , for all places i ∈ P ;
φ(M)(i) =
µ, i=structure.
ψ(R) = ⋃
ψ(r i ) with ψ(r i ) = t i .
r i∈R
, and
Proof. The function φ represents a bijection between the multisets of objects
of Π and the markings of CP N Π based on the corresponding links between objects
and tokens, and between membranes and places, respectively. Let (w 1 , . . . ,
w k , µ) be the multisets of objects from the membrane configuration M, together
with its structure µ. Similarly, for a set of rules R ′ = {r 1 , . . . , r i } of Π, the
function ψ is a bijection constructing the set ψ(R ′ ) = {t 1 , . . . , t i } of transitions
of CP N Π from the set R of rules.
A membrane configuration M 1 can evolve to a membrane configuration M 2 by
applying an evolution rule r from R ′ if and only if, given the marking φ(M 1 ), one
obtains the marking φ(M 2 ) by firing a transition t in CP N Π , where ψ(R ′ )(t) = r.
Overall, this is a direct consequence of the fact that ψ and φ are bijections. ⊓⊔
From the construction above, it results that the initial configuration of Π
corresponds through φ to the initial marking of CP N Π . Moreover, according to
Theorem 1, it results that the computation of Π coincides with the computation
of the CP N Π .
6 Simulating LDL Degradation by Using CPN Tools
Now the LDL degradation pathway description by using mobile membranes could
be encoded in colored Petri nets. Such an encoding is provided to allow the use
of a complex software tool able to verify automatically some important behavioral
properties. Some decidability results for behavioral properties of membrane
systems with peripheral proteins are presented in [5], but they cannot be proven
automatically. For colored Petri nets is available a complex software called CPN
Tools in which simulations can be performed, and certain behavioral properties
can be checked automatically: reachability, boundedness, deadlock, liveness,
fairness. CPN Tools (www/cs/au.dk/CPNTools) is a tool for editing, simulating,
state space analysis, and performance analysis of systems described as colored
Petri nets.
In what follows we show how the rules of mobile membranes used to model
the LDL degradation pathway can be simulated using the CPN Tools. To make
easier to observe how the evolution takes place using CPN Tools, we simplify
the system and use only the transitions that eventually occur.
A CPN model is always created in CPN Tools as a graphical drawing. Figure 3
describes the LDL degradation pathway model, namely the membrane configuration
M 1 from Section 3. The diagram contains eight places, four substitution
transitions (drawn as double-rectangular boxes), a number of directed arcs connecting
places and transitions, and finally some textual inscriptions next to the
places, transitions and arcs. The arc expressions are built from variables, constants,
operators, and functions. When all variables in an expression are bound
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